A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a \(P_5\)-free graph with clique number \(\omega \ge 3\) has chromatic number at most \(\omega ^{\log _2(\omega )}\). The best previous result was an exponential upper bound \((5/27)3^{\omega }\), due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for \(P_5\), which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for \(P_5\)-free graphs, and our result is an attempt to approach that.

中文翻译：

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色数的多项式界。IV：排除五顶点路径的近多项式界

如果图G没有与H同构的导出子图，则它是H无的。我们证明，团数为\(\omega \ge 3\)的无\(P_5\)图的色数最多为\(\omega ^{\log _2(\omega )}\)。由于 Esperet、Lemoine、Maffray 和 Morel，先前的最佳结果是指数上限\((5/27)3^{\omega }\) 。多项式界限意味着著名的 Erdős-Hajnal 猜想适用于\(P_5\)，这是最小的开放情况。因此，人们对无\(P_5\)图是否存在多项式界限非常感兴趣，我们的结果是尝试接近这一点。